After playing through the first Contract ("The First Hit") properly, I noticed that the only requirements were to kill the target. It didn't seem to matter if you got spotted or anything else, so I thought - that would be perfect for a Speedrun!
So, I started up Bandicam and got down to it:
Level: Hitman:Absolution - Contracts ("The First Hit")
Time: 21 seconds
Runs: About 4-5
After a couple runs of seeing what the fastest path might be (leaping over the log takes more time than running around imo), I changed to a smaller resolution and turned the graphics down to low settings to help with speed. I don't know if this is the fastest possible route, I just found it worked well instinctively after a few runs.
Bandicam seems to have some trouble with H:A, the fps keeps popping through, and seems to lag it out a bit. I even tried disabling SLI. I thought it might be my NVIDIA cards, and in the end, I ended up running it on an AMD/ATi HD 6870 even smoother than dual GTX 560Ti's (this game prefers AMD videocards).
Good luck with it and See You In The Games!